Compendium of Chinese-Rings-Like Puzzles

There are some puzzles that have been there for a long time and have their roots that far back in the past that their origin is not known exactly. The traditional six piece burrs are one such example, the Chinese Rings puzzle another. For some information about the history of the Chinese Rings (also known as "Nine Linked Rings") see [9].

In this compendium, we exhibit a collection of all puzzles known to us that are all related to this type of puzzles, why we will call them "Chinese-Rings-like puzzles", or more formally, "CR recursive puzzles". A common similarity for all these puzzles is that they have an astonishing high number of moves to solve and they have several rings or pieces working in a similar way in the solution.
Our Compendium features a large variety of different puzzles: Disentanglement puzzles, sliding pieces puzzles, puzzle boxes, burrs, sequential movement puzzles, ball maze puzzles, and even puzzle games with certain rules.

The last example above, the "Kugellager", is the puzzle that started our research and lead to the first article about this puzzle and similar puzzles: [1]. In that article, the "Kugellager" puzzle with its 1250 moves is analyzed and later on short analyses of other puzzles like "Die Welle" were added and a table was created showing several puzzles that are now in this compendium. After collecting some of the related puzzles, a group page was set up and the term "n-ary puzzle" was termed for these puzzles: [2]. This term n-ary was also used in a nice article [11] about designing n-ary puzzles which appeared in CFF 15 in November 2014. The most recent exteded version of this article can be found in [12].

The list of puzzles is the main part of this compendium. It lists the puzzles with several details in entries ordered alphabetically by their names.

The first three lines of each entry contain some general information: Name, designer, manufacturer and year of release.

In the next two lines some parameters relating to the mathematical classification of the puzzle are provided: The arity (usually denoted by "n" in this compendium), type and number of special pieces (denoted by "m" in this compendium),

Then an explicit representation of the solution length function (if known) and the number of moves of the particular puzzle with values n and m -- either counted or derived from the solution length function.

Each entry also has a unique puzzle-ID, to simplify puzzle reference. These IDs do not follow the order of the puzzles in the list, as puzzles entered into the list later on will have greater IDs, but might occur early in the list because of their name in the alphabetical order.

Links to patents, Extremely Puzzling page, other web pages on this puzzle

Symbols:

§=counting moves of special pieces only; ‡=counting moves until first piece comes out

Record Layout — Structure of each puzzle entry in the list

Note on the Arity: While n (and hence the arity) by Definition 3.1 in this compendium is the number of different positions each piece can have (including possibly the "removed" state if it occurs regularly during the n-ary sequence, not only at the end when all pieces are removed at once), there are puzzles, for which a generally accepted arity has been defined by the designer of the puzzle, and which is usually aligned to the solution length, and the base of the exponential function describing the solution lengths. In this cases, a bracket notation is used in the arity field for the defined arity n' and the number of levels n:
n' [ n ]
For example, the Racktangle puzzle would have defined arity 3, yet the pieces have 5 different positions (two additional positions to ensure the solution length proportional to 3m), and this would be denoted by: "3 [5]".
If there are single pieces with different number of positions (e.g. when only one piece comes out of the puzzle in the n-ary sequence), then only the number of positions/levels for the majority is provided.

Hint: Even if the list cannot be ordered for other fields, you can easily search for a certain entry using your web browser's built in search function (usually invoked by keys Ctrl-F or F3).

Hint: For previews of secondary images or the variants (entries with IDs), please hover the mouse pointer over the corresponding link (without clicking). In the top right corner of the browser window, a preview will appear. Unfortunately, this does not work on all browsers; in particular some Internet Explorer versions do not show these previews.

Beside this introduction and the list of puzzles there are also a page with a formal definition of the structure of puzzles that belong into this compendium, including some properties and examples. The compendium is concluded by a page to contribute to this compendium by adding puzzles or new information. Please have a look at the navigation bar on top of all pages or the table of contents below.

The Algorithme series features different puzzles with different number of discs, different disc heights and post heights. They are all some variation of Tower of Hanoi, which can also be seen in the rules: move one disc at a time, which is on top of its pile; no bigger disc may be put on a smaller one (equal size is OK); piles may only go up to post end, not higher.

Variant of CR160, but not coming apart and only with 5 binary pieces, and one lid piece to be removed by the solution sequence. The lid piece has a different structure, and leads to four puzzles: Alken has lid piece which is binary (135 moves) in one orientation and 6-ary (321 moves) in the other. Kenal has a binary (135 moves) and 4-ary (257 moves) lid piece. The 135 move configurations also allow the solution sequence to run over the point branching into the last few moves before lid removal, and then leading to a dead end. These dead ends can also be reached when re-inserting the lid and trying to close the box. Also at the beginning of the 135 move sequences (box closed) there are some dead ends possible. Second pictures shows the box open and details of the lid pieces, the third piece shows the lid to be slid open without removal, possible for the Kenal 135 configuration just before the end of the sequence.

Goal is to remove the big foldable circle piece, which starts in the middle. It can traverse to the left or right end, both consisting of 5 ring+connector pairs. While there are two states for the big circle and each ring+connector pair (i.e. binary), the overall solution is linear. Each pair is traversed only once. The zig-zag chain of rings on the main loop looks like the structures used in others of Aaaron's n-ary puzzles.

3D printed reproduction of CR039 by original designer; variant:CR137. Each of the six pieces is assembled from three 3D printed pieces (one silver, two black), and some screws. The second picture and second reference show the add-on to the "Master Set", which contains an additional set of body pieces, so that two full cubes can be built in parallel, an extra bronze colored piece for keeping track of the orientation, and additional inserts, so that the following puzzles can be built: Barcode burr (black, binary, by Lee Krasnow), TernCode Burr (orange, ternary, level 115, by Derek Bosch), QuadCode Burr (yellow, quarternary, level 1233, by Derek Bosch), SuperCode Burr (red, level 81.38.11.11.6, by Lee Krasnow), ExtremeTortureCode Burr (white, red, orange, level 139.6.1.17.6, by Lee Krasnow and Derek Bosch), CoordiCodeBurr (blue, coordinate motion and binary mixed, level 7.5.3.4.1, by Lee Krasnow). The third picture shows the paperwork coming with this puzzle, including some overview, detail cards for each puzzle, an assembly guide, a hint and solution guides, solution (Grey code printed in shades of grey), and diagram plans that can be used to keep track of the maze positions during the solution, for which some small nuts are included as markers. In the beginning of 2019, some more inserts were designed by Lee for his BarcodeBurr, but with a focus on coordinate motion and shorter, less regular solutions, not the long n-ary sequences. This set can be seen in the fourth picture and third reference.

This is a combination of puzzle box and burr. Not only the panels can be opened and removed, but also the frame can be taken apart completely. Inside the box is a second puzzle, the Reactor by Eric Fuller, a small puzzle box.

V1, N01; Two interlocking binary sequences (one of bars, one a bit hidden of the sticks). Beside the binary moves, this puzzle also contains burr moves without an n-ary scheme and with half-notches. The number of moves contains the binary sequences and some of the burr like moves.

Additional mechanisms, modified sequence, combination of several binary sequences. The objective to remove the dice modifies the sequence even further, as a die can only be taken out when a drawer is fully extended and the drawer above in its starting position inside the box.

The mechanism is hidden and seems to consist of the four sliders, several ball bearings, and sliding pieces. There is also one additional ball bearing that has to travel from start to goal, from where it can be put into the start position via a reset feature. During this time, the ternary sequence is executed twice (forwards, then backwards) with 54 slider moves each. The total number of moves includes these slider moves (2·54), the corresponding tilting moves to move ball bearings/sliding pieces (2·54), tilting moves to move the extra ball bearing inside the puzzle and out (3+1). Once the extra ball bearing has reached the half way position, it can go inside the sliders and cause some lockups that have to be undone by reversed moves before the regular sequence can continue.

Complimentary combination of several different sequences: Top 5 bars run in a 3-ary sequence, together with the 5 bottom bars, who run (slower) in a 2-ary sequence. These interact with the 4 loop-pieces, which run accross in a 3-ary sequence. First challenge of the puzzle is to understand these sequences, then the second is to disassemble and correctly reassemble the puzzle, with many other parts, alltogether 29 pieces.

Variants: CR178, CR188. This is mainly a (binary) Chinese Rings puzzle with single rings. The second ring of each pair is dropped from the main bar when the corresponding ring get's off the bar. It will then stay unhooked, while the primary ring follows the usual Chinese Rings sequence. For each pair there are four states (on/off loop for each ring), so this puzzle can also be considered quarternary. However, the main sequences and interactions are only binary, with touching every secondary ring only once, hence classified as binary here.

Variants: CR175, CR188. This is mainly a (binary) Chinese Rings puzzle with single rings in a chinese rings chain (CRC) and then an additional chain, a zig zag chain (ZZC) through the connector piece ends. When the last ring from the CRC on the handle bar is dropped, a sequence through the ZZC follows. As this is a ZZC, half of the rings are wrongly oriented for the usual sequence, and at those points parts of the CRC are traversed to the beginning of the CRC, to allow access to the ZZC rings in the other orientation. These interruptions in the ZZC sequence by CRC sequences will then happen until the completion of the solution. The main scheme is that the rings of the CRC come off one after another like in a Chines Rings puzzle. Consequently, the reassemlby follows this scheme: Run through the CRC to put on the last free ring of the CRC, then put the lasts free ring of the ZZC on the handle bar. This automatically adds two rings of the ZZC, so one will need to be released to allow to put on the next CRC ring. For this some ZZC sequences are required, with some CRC sequences performed up to the correct entry point of the ZZC. The scheme can be learned with a few ring pairs (up to 4) initially, but only with 7 or 9 ring pairs, all required moves become apparent.

Variants: CR175, CR178. Like the first two puzzles in the series, the CotC III is mainly a binary chinese rings puzzle. Each ring is part of a pair with a free ring (only one end caught in a connector) and a ring part of the main zig-zag back bone. After analysis, the puzzle can be solved with some simple rules: Each ring has only one correct orientation on the handlebar piece. The free rings form a binary chinese rings puzzle that needs to be solved, and when the bar needs to go through one ring of a pair, it should always go through the free ring. The last rule is about re-assembly (entanglement): When the handlebar is at the rightmost free/zig-zag ring pair, it should break the rule before and go through the zig-zag ring.

Goal is to compress the digit stack to minimal height by rotating discs and moving them verticalle, and additionally to line up the red markings with the four red markings on top and bottom parts. There are several different-length dead end sequences. The five gray discs move in a binary symmetric Gray code sequence, unlike the black ones. Each disc has an orange pin, which can interock with two different holes in the disc below, i.e. two differet positions for each disc.

While the rings in the classic Chinese Rings puzzle are linking their connector with the next connector each, in this one, the regular scheme is broken and some rings go over the next two or three connectors. Some of them lead to irregularly stacked rings on some connectors, while for others the rings over the next one and two connectors are aligned in parallel over one connector. When solving, one has to ensure to choose the right ring for the sequence and which ring to skip, while the overall solution sequence is aligned to the general Chinese Rings sequence.

One of the simplest designs of a whole puzzle family, with different number of sliders, disks, and arities. This design was devised fist for higher order variants in August 2014, about a month before this puzzle. One of the higher order variants is CR149.

The main chain of this is a classic 9 ring Chinese Rings puzzle. Attached to rings 3, 5, 7, and 9 are a small ring and connected to that two regular sized rings. Those rings are linked with the previous and next connector. During the solution, only at most one ring of each additional ring pairs will be on the handle. The main solution sequence is still binary, but one has to determine when to pick up the forward / backward secondary ring. At some points in the solution, both the primary ring and the secondary ring are on the main bar, at other points in the solution, also only the secondary ring might be on the main bar (but this only holds for the forward rings, the secondary backwards rings are never on the handle alone). Therefore, the puzzle could also be classed as a ternary puzzle, or even quarternary, but the main structure is still binary.

This is actually a section of an infinite puzzle: The puzzle could be extended infinitely to the right or left. If this structure is closed as a loop, this will lead to something like CR197. Therefore, the arity is hard to determine. There are six loops and the configuration off the loop for each pole/sector, so 7-ary might be a good description. However, the solution only makes use of 4 of the poles (and the rope off the puzzle), so it is more 5-ary, and the actual solution length is 3-ary. For the solution the two bends of the rope start in the compartments denoted by red triangles, and each bend will be maneuvered off the puzzle separately, with 40 moves each. In the IPP38 Design Competition it participated as part of "Loopy Lattice Puzzles"; other puzzles from the same series: CR196, CR197

The three states of each ring+loop pair are: main bar through the loop (or "fishing hook", initial configuration), through the ring, and off both. When reassembling the puzzle, an additional challenge arises: it may easily happen that some hooks end up on the main loop in wrong orientation. As this can only be seen after many (up to 1000s) of moves, careful planning is advised and analaysis of smallers problem of the first few hooks only. One feasible approach is to arrange the loops in an alternating pattern above and below the the backbone while running through the sequence.

The puzzle can be built with various slider shapes, leading to different mazes. Most of them are not n-ary, like the one shown in the pictures. Several puzzles have been implemented as online version (see reference [2]), an n-ary one has also has been implemented — please see reference [3], and for this the solution length and other details are provided here.

Goal is to move all light sliders down and all dark sliders left. By unlocking and removing the transparent lid, all little square pieces can be reoriented, allowing for 425≅1015 different challenges. Not all of these are possible as can be seen from the second picture, where a partial configuration is shown with the two top-left slider pairs blocking each other, unable to move. While the first picture shows the simple standard configuration of the puzzle, the third one shows one adapted from the N522 puzzle (CR087), with nontrivial solution and polynomial solution length. The letters of the name depict the various configurations of the small squares.

Beside the 8 main sliders, the puzzle contains several other smaller sliders for the interaction between the 8 main sliders. Additionally, there is a small ball and a ball maze in this puzzle, and the goal is to get the ball out at one of the three maze exits. The maze is also part of the sliders (see second image) and therefore the binary character only holds for the basic puzzle, without the ball.

Lego variant/implementation of CR010. Second picture shows the three different piece states, with one moved out to the right already. Second reference links to building instructions created by Jeremy Rayner; the puzzle can be built with the pieces of a Mindstorms NXT set, but slight modifications might be necessary depending on the actual piece set.

Variants: CR048, CR056. Mechanism is completely made out of wood, no metal (pins) used. Kamei also included a second alternate solution with a shortcut, which will only work at the beginning of the usual sequence, and is a couple of moves only.

First goal is to remove the coin, second the whole thread from the metal part. Both challenges are the same ternary puzzle repeated, but for releasing the coin additional restrictions exist. This seems to be a variant of the Meiro Maze shown in reference 2.

Variants: CR126, CR136; This puzzle is a further developed variant of the original Power Tower, and as such it also comes as a whole set of pieces. With these pieces coming as 2-ary, 3-ary, 4-ary, 5-ary, and 6-ary (in the version shown in the pictures), different configurations can be created. There is a special binary piece as a key piece that is part of all configurations as top piece. Therefore, there are 6 slots and 5 of each piece arity (only 2 for 6-ary). Reducing the massive block to a slim tower allows pieces of different length and theoretically in arbitary arity without changing the central tower or other pieces. In the pictures, different examples are shown: 3 binary pieces (solved), 6 binary pieces (solved), one of each kind (mid-solution). While the Power Tower has pairs of mirror-symmetric pieces, here all pieces of same arity are the same and have to be entered in a helical pattern. While the sequences for even and odd arity pieces differ especially at the beginning, they are the same in this puzzle. Goal is to choose a configuration, enter the pieces into the tower, and slide them until they are all flush with the tower side on one end. The maximum number of moves for the puzzle in the picture is 13432, with pieces: (2*, 5, 5, 5, 6, 6).

One of a whole puzzle family, with different number of sliders, disks, and arities. CR143 is a simpler variant. The goal of MinTer-MaxTer is to move the sliders from the outer discs with two slots to the outer disc with 8 slots and collect them there.

Main binary chinese rings chain, with three additional binary chains of 2 rings each, attached to rings 5, 7, and 9. These are interwoven with the main chain, leading to ternay subsequences, with some quaternary positions, where two subchains meet. Of each of those additional sequences, there is always only one of the two rings on the main loop. This is one of six puzzles in the Chinese 99-ring series.

This one is based on a main binary chinese rings chain, with an additional chain of 3 rings starting under the 5th and 8th ring. Unlike CR171, these additional chains are not rings linked directly with each other, but each ring is connected to one of the vertical bars via a smaller ring. In the starting position, these look like linked chains, during the solve, the chains act like secondary chinese rings chains, and therefore also multiple rings of the same secondary chain will be on the main bar at the same time, especially when one of them is put on/off the main bar. While the overall structure of the chains is binary, each of the four possibilities for each ring pair of ring on/off the handle (on/on, on/off, off/on, off/off) occurs and this puzzle could also partially be classified as a quarternary puzzle. For each pair of primary and secondary ring, putting on/off each of the rings of the pair requires a traversal of the lower rings sequence, making it a quite long solution sequence, adding up all these binary sequences. From the solution standpoint it might therefore also be classified as being partially ternary, and probably this is the main influence on the solution length. This is a later puzzle of the Chinese 99-ring series.

Variants: CR087, CR132,CR133; Beside ternary sliders in two orientations, there is a simple ball maze built into the left end of the sliders and the ball is to move from bottom to top as goal, while the moving sliders obstruct and open some of the maze parts. The second picture shows the position in which the maze is usable; only the first slider has to be moved up and down while the ball traverses the maze. The number of moves is for putting all sliders up/to the right (calculcated with Burr-Tools, see second reference), with one additional move of the left slider to remove the ball, totalling 21.

Variants: CR087, CR131,CR133; Beside ternary sliders in two orientations, there is a simple ball maze built into the left end of the sliders and the ball is to move from bottom to top as goal, while the moving sliders obstruct and open some of the maze parts. The second picture shows the position in which the maze is usable; only the first slider has to be moved up and down while the ball traverses the maze. The number of moves is for putting all sliders up/to the right (calculcated with Burr-Tools, see second reference), with three additional moves of the left slider to remove the ball, totalling 71.

AKA: "522"; Variants: CR131, CR132,CR133; Beside ternary sliders in two orientations, there is a simple ball maze built into the left end of the sliders and the ball is to move from bottom to top as goal, while the moving sliders obstruct and open some of the maze parts. The second picture shows the position in which the maze is usable; only the first slider has to be moved up and down while the ball traverses the maze. The number of moves is for putting all sliders up/to the right (calculcated with Burr-Tools, see second reference), with five additional moves of the left slider to remove the ball, totalling 217. This is the first model of the series. Physically built have been all versions from 2+2 to 10+10 sliders, and some are presented on this page.

Variants: CR087, CR131,CR133; Beside ternary sliders in two orientations, there is a simple ball maze built into the left end of the sliders and the ball is to move from bottom to top as goal, while the moving sliders obstruct and open some of the maze parts. The second picture shows the position in which the maze is usable; only the first slider has to be moved up and down while the ball traverses the maze. Number of moves yet to be determined. This is the biggest of the series actually built.

Starting position is with all green arrows pointing to the green rectangle (right). Goal is to turn all the elements on the modules with the blue arrows pointing right. There are dead ends and it is not always immediately obvious which move should be the next one. The modules have arities (left to right): 4,3,3,2,3,2. The puzzle won a Jury First Prize in the 2017 IPP Nob Yoshigahara Puzzle Design Competition. The name is based on the following little anecdote relating to the movement of the pieces: The teacher-choreographer of ballet school gathered the students of various classes near the bar and tried to arrange a new divertissement with pirouette as the main element of group dance in a limited area. He ordered to make pirouette one by one to avoid collisions. But it was impossible in limited space to do that. The lesson failed. Somebody suggested asking the math teacher to help. Luckily the mathematics was passing near and was interested in assigned task. After some measuring, he proposed a scenario and exclaimed: One by one spin back and forth and no full pirouette!

Variants: CR125 and CR139. The first piece shows the puzzle on the stand with 9 sliders entered and one of each arity 3, 5, 7, and 9 coming with the set, and some sheets listing number of moves for certain configurations. Those sheets are based on above formula involving the base/arity nm of the leftmost piece and the product n0··· nm—1 of the arities of the other pieces, regardless of their order. The second picture shows the puzzle with 9 sliders and all 16 knobs in two rows. The third picture shows all pieces of the set, including the leftmost pieces (called "starting block") for each arity, and the common piece. There are following piece counts: 3-ary: 7+1 (1 block attached), 5-ary: 4+1 (2 blocks), 7-ary: 3+1 (3 blocks), 9-ary: 3+1 (1 blocks), 1 common piece, 16 knobs. The reset and piece number selection mechanism has not been shown in the pictures, as finding this is an extra puzzle posed by Johan.

Variant of CR035 which includes overlapping and interacting Panex instances. As the original design was not solvable (as discovered by Bob Hearn), this puzzle has to be modified by removing the blocking mechanism in the center. Goals are: 1) to swap pieces horizontally (e.g. A and B), and 2) swap pieces vertically (e.g. A and C), obeying the Panex rules, i.e.: in the vertical grooves, no smaller piece can be below (i.e. closer to the center) than a larger piece, same for the horizontal grooves (no larger piece closer to the center, but for both sides of the groove). This modification was proposed by Diniar Namdarian in 2015. The solutions provided have 46 moves for swapping A and B, and 68 moves for swapping A and C.

Binary sequence, which is non-GC based and uses a pin-maze-mechanism, a little trick was added corrupting the sequence and making it more interesting for the solver. The third picture shows the prototype, which has a simpler frame but same sequence, the last picture shows both puzzles.

Variant: CR035, CR009. This puzzle is basically a Panex puzzle with only 3 levels (6 pieces) instead of the 10 levels (20 pieces) of the Panex puzzles. Goal is to slide the pieces (without turning) to exchange the yellow and blue stacks.

Variants: CR126, CR167; This Power Tower is a whole set with a block hosting up to 6 stages, a blocker piece to set the number of stages (between 3 and 5, 6 stages without blocker), and a set of pieces for each of the two orientations (two different woods). The pieces come in binary, ternary, and quaternary shape and can be combined arbitrarily, leading to mixed (or uniform) base sequences, which can be quite confusing. There are 1080 different possibilities, with the level varying from 11 to 2724. The solution length is for a uniform n-ary configuration with m pieces. Addition: This now includes an extension set of quinary pieces. The overall entry now contains these pieces and there are now solutions possible up to level 7806.The second picture shows this extension set.

This puzzle contains multiple challenges, i.e. starting configurations of the loop: adjacent compartments or opposite compartments. The whole puzzle consists of four modules/sectors with four loops each. During the solution sequence, the loop bend will traverse through multiple sectors/modules. The solution works in several stages: First, remove one of the bends from the puzzle (steps 1 to 41), and then only one bend will be caught in the puzzle center. Then in the second stage, remove the other bend, which may be accomplished in several ways. Move the free bend into the puzzle via a different path so that both bends meet at the end and the rope can be pulled out, or remove the second bend like the first one before. The configuration vector will denote the position of the rope (bend) in the compartments defined by the layers of loops, counting them from innermost to outermost.During the solution, the rope will go through at most one of them at each time, leading to configurations of 0 to 4 (number of sectors/modules). The solution does not make use of all combinations, leading to a ternary solution path length. Other puzzles from the same series: CR196, CR198

Variable number of stages (1 to 4, box is built modular) and plates of base 2 and 3 included, which together with the solid plate for the lowest position, can be used to create all mixed base 2 and 3 puzzles for up to 4 stages.

Variation directly created from Chinese Rings by attaching a small connector ring and a second bigger ring to each ring, below the first and around the same vertical rod. There are four states for each ring pair: main bar through lower ring (initial position), through upper ring, through both rings ("double ring"), and off the rings. All those appear in the solution, and the double ring configuration is used to mimic the classic binary chinese rings. The configurations with one ring on the loop appear exactly once in the solution sequence, and their transitions interrupt the binary sequence in a regular pattern and increase the number of moves considerably. This is one of six puzzles in the Chinese 99-ring series.

This puzzle looks like a chinese rings with all the rings put on the main handle backwards. At a closer look, each ring has a second ring attached at the bottom. To solve this puzzle, the bottom chain has to be solved like a standard chinese rings puzzle, and at the end of each run, one more ring from the reversed top chain comes off. While there are more than two states for each ring pair (4 states, each ring can be on or off the handle), the main sequence is binary, which is why it is classed binary here, and considered as chinese rings puzzle with some extensions.

While the rings in the classic Chinese Rings puzzle are linking their connector with the next connector each, in this one, each ring links its connector with the next two adjacent connectors. The solution is based on the Chinese Rings solution, and is in fact the same sequence like for the Dispersed GC lock CR074

The cube in the picture is a version where only the first piece can be removed, called the "136 Minutes Cube". It comes with an alternate piece, which can be used to raise the number of moves to 12282 moves, calles the "206 Minutes Cube&quot. These names refer to an estimate of solving the respective puzzles.

Each metal ball can be in a top-left, bottom-left, or a bottom-right position, and there are corresponding slider positions middle and top. The bottom slider position occurs only during transition of ball between left and right. A newer circular version replacing balls by switches is: CR083

The second picture shows the different position of the special pieces, the pairs of yellow blocks in positions 0, 1, and 2. There are other positions not part of the solution. Recently, we found a shorter, non-ternary solution that was not intended, with goal configuration in third picture; under investigation.

There exists an unintended shortcut solution with 49 moves (see Jaaps's page below). The second picture shows an unknown mini variant with rule scales in cm and inch, and inscription "PAT NO 23596" on the back side; Variants: CR022, CR026

Round variant of CR064. Goal: Move all handles to the outer position and reveal hidden message "Nicht durchdrehen", German for "do not get mad" and also referring to turning the steering wheel (German: Steuerrad). The second picture and reference show a box newly released in 2018, which features the same puzzle as lid. To open the box, all the sliders but the short one have to be moved to the outer position. This makes the solution shorter than the one of the original puzzle.

Smaller version of original Super-CUBI with adjacent panels moving on opposite sites. Comes with a solution leaflet showing all 324 moves, and additionally some instructions on how to calculate and identify the current configuration. Varaiant: CR025

Move count includes control bar; the first picture shows the whole group of the Binary Burrs (small) with 3 to 10 special pieces, all with solid cage, the other pictures show the individual puzzles; Variants: CR076, CR012

While the rings in the classic Chinese Rings puzzle are linking their connector with the next connector each, in this one, each ring links its connector with the next three adjacent connectors. This is a logical extension of CR190.

extra rings for symmetry; third picture shows puzzle Dingo Trap, a variant with the rings separated and held by smaller loops; reference [14] shows this variant including building instructions and solution

Mechanism similar to CR084. Goal: Move the one ball with the special starting position to its third hole and remove (only) this ball from puzzle. The two pictures show second (more stable) and first edition.

The basic mechanism of the box is a ternary mechanism consisting of discs with the switches attached and visible to the puzzler, and some ball bearings. These will move into some cutouts of the discs and block them in various positions, same general concept as in CR064. Additionally, there are two mechanisms interacting with several discs each: One visible as the bottom horizontal slider, the other hidden, but with its state visible through a small hole below the left disc. This mechanism and the ball bearings have to be manipulated via tilting. The lid contains the mechanism and is firmly closed. However, a second variant was released with a transparent top, allowing the puzzler to see most of the mechanism.

A series of boxes with arity 2, 3, 4, and 5 was built and this is the highest arity one. All boxes have 3 drawers and two plates for the top mechanism, and a main drawer to open after the sequence has been completed. The models differ in their acrylic plates, which are engraved with a label stating their arity.

This puzzle is based on: CR057. Variants: CR184 and CR185. The second picure shows the goal configuration (all sliders moved to the border), and the third picture a different colour variant in a configuration during the solution.

This puzzle is based on: CR057. Variants: CR183 and CR185. The second picure shows the goal configuration (all sliders moved to the border), and the third picture a different colour variant in a configuration during the solution.

This puzzle is based on: CR057. Variants: CR183 and CR184. The second picure shows the goal configuration (all sliders moved to the border), and the third picture a different colour variant in a configuration during the solution. The solution length function for the number of moves f(m) is a solution of the recursion f(m) = 3f(m-1) + 4f(m-2) +3 with f(0)=0 and f(1)=1 derived from solving the puzzle with m pieces.

This compendium contains a special collection of puzzles which are somehow related to the famous Chinese Rings puzzle. For a definition which puzzles are included in this compendium and for a clear understanding why they are included, we will provide a structural definition of the class of puzzles in this compendium, the CR recursive puzzles. We will provide a definition, with some examples, and with some interesting properties of these puzzles.

A CR recursive puzzle is a puzzle that contains m special similar pieces (with m ≥ 1) and

the puzzle can be generalized to other values of m ≥ 1 and

each special piece has n different positions (e.g. 0,...,n—1, with n ≥ 2) and

there is a uniform condition stating that a special piece can only move between some positions if the other special pieces are in certain positions.

Note that beside the special pieces in the definition above, there may be other pieces. For a distinction from the others, the special pieces are sometimes called "ring pieces", in analogy to the classic Chinese Rings puzzle. Also note, that one of the different positions each piece could have is the "removed" state, when it is extracted from the puzzle. This state is only counted in this definition if it occurs within the n-ary sequence and occurs regularly, not when at the end all pieces are extracted from the puzzle one after the other.

This is a short and formal definition related to the structure of the puzzle. Some examples might be useful.

In the picture, a typical wooden version of CR with 5 rings is shown. Each ring may be positioned (diagonally) on the horizontal loop or off the loop. These states may be denoted as: 1 - on, 0 - off the loop, so in this example we have:

m=5, the puzzle contains 5 rings and

n=2, the puzzle is binary

and the uniform condition for moving a ring is:
The k-th ring can be moved between 1 and 0, if and only if ring (k—1) is in position 1 and all rings to the left of (k—1) are in position 0.

To conclude the matching of the particular classic CR puzzle of this example with our definition of a CR recursive puzzle, we note that there exist many different classic CR versions, with various number of rings, that correspond to the different values of m in our definition of CR recursive puzzles.

In this puzzle, we have a slider carrying a line of discs. Each disc can be either in a vertical position (denoted by 0) or horizontal position (denoted by 1). This puzzle is equivalent to CR, if we restrict the moves to the ones that are used in the solution: From the vertical position, each disc can be turned left into the horizontal position, or to the right to a different horizontal position. We disregard this (right turned) position, as it is not needed for solving the puzzle. To see the uniform condition for this puzzle, we have a look at the pictures above and note: A disc can be turned between 0 and 1, if the disc immediately to the right is 0 (in vertical position) and all discs further right are 1 (in horizontal position). For the solution, two additional restrictions are implemented: Only a disc at the position with the additional space (second from the right) can be turned. Discs can only be moved out to the right when in position 1 (horizontal).

The Crazy Elephant Dance is a generalized version of SpinOut. Instead of discs, we have a line of 5 elephants on a slider, and each elephant has three possible states: 0 - facing upwards, 1 - facing to the right, and 2 - facing downwards.

The uniform condition is split into two parts in this case:
1. An elephant (the second from the right in the picture) may move between 0 and 1, if the one immediately to the right is in position 2, and (not shown in the picture) all further right are in position 2.

2. An elephant (again the second one) may move between 1 and 2, if the one immediately to the right is in position 0, and (not shown) all further right are in position 2.

Again, as for the SpinOut, the second part of the conditions above arises from the fact that the slider and elephants may only move out to the right when in position 2.

The pictures demonstrating the two parts of the condition also show an example how to move the second elephant from 0 to 2: It first has to be moved from 0 to 1 (first two pictures), then the elephant to the right of it is moved from 2 to 0 (third picture), then the second elephant may finally move from 1 to 2. This gives an idea what is necessary to move the leftmost elephant from 0 to 2, which is a vital step in the solution sequence.

The Kugellager has four balls, which are the special pieces in our definition and one slider containing some mazes for these balls. The four balls move up and down (maze in the slider permitting, green arrows), and the slider moves left and right (blue arrow). Each of the balls has 5 regular positions {0,1,2,3,4} (or {1,2,3,4,5} in the picture), and then there is an additional position 5 (or 6 in the picture), which can only be used if all balls are already in position 4, to remove the slider afterwards. This is why we may disregard this position 5.

The pictures are taken from the article [1] which also describes the movement in more detail and also lists the uniform condition for ball movement. Shortly summarized, a ball may move between positions i and (i+1), if all balls to the left are in a certain set of positions and all balls to the right are in a certain (different) set of positions.

This puzzle can not only be generalized to more balls, as the definition requires, but also to a higher or lower level, as the Kugellager 7 puzzle (n=7), or the Auf dem Holzweg puzzle (n=3) show.

At first sight, Tower of Hanoi looks very different from the puzzles we have seen so far, but it is a CR recursive puzzle and complies to our definition: It has m discs (m=9 in the picture), and each of the discs can be on one of the three piles built on the poles, so n=3. There are different variants with different numbers of discs, so the generalization to other values of m (even other values of n with more poles) is easy. All these variants will have to obey simple rules: Move only one disc at a time, and only the top disc of a tower, and a disc may only be laid down on discs that are bigger than itself. The rules deliver the uniform condition we need for our CR recursive definition and can also be translated to: To move a disc, all smaller discs have to be on a pole different from the start and destination positions of the move.

The uniform definition covers a property that is independent of the actual number of pieces and also allows the same condition to apply to each one of them. This condition states that for all suitable indices i and j, piece number i can move between positions j and j+1 if and only if the "lower" pieces (left of i) and "higher" pieces (right of i) are in certain positions. This works independently of i: No matter if the second piece (i=2) or the fifth piece (i=5) is to be moved.

There are also puzzles for which the "higher" pieces are irrelevant, for example:

SpinOut: In the example above we have seen that to toggle a piece (vertical ↔ horizontal), the piece one below must be vertical and all below that must be horizontal. This is the uniform condition for the lower pieces, the higher ones do not matter.

Kugellager: In article [1] the conditions for ball movement are sketched and depend on the source and destination positions of the current move. We just take out one example of the conditions in this article. It states, if a ball moves between positions 1 and 2, all lower balls have to be in position 5, and all higher balls have to be in position 1, 2, or 5. This rule holds for all balls, so it is a uniform condition.

All puzzles contained in this compendium allow a recursive description of their solution. As an example, take the Tower of Hanoi. This puzzle has three positions, two of which are initially empty and one carries a stack of discs that are ordered from biggest in the bottom to smallest on top. The aim of the game is to move the stack of discs to the third position (whichever that may be) obeying the following two rules:

Only one disc may be moved at a time

Never place a bigger disc on top of a smaller one

A typical Algorithm to solve this problem can be described informally as follows:

Move Tower of n discs from startTower to endTower:

6 - startTower - endTower → auxiliaryTower// the tower that is not one of the two above

if n>1 then

Move Tower of n—1 discs from startTower to auxiliaryTower

Move one disc from startTower to endTower

if n>1 then

Move Tower of n—1 discs from auxiliaryTower to endTower

This algorithm works by moving the n—1 top discs away on the auxiliary tower, then disc n, then the ones on the auxiliary tower to the destination tower.

What algorithm do we use in lines 3 and 6 in order to move an n—1 disc tower? It is our very same algorithm, that calls itself recursively, and now we have our justification to call this puzzle "recursive", as it can be solved by such a recursive algorithm.
More details about Tower of Hanoi can be found e.g. here: [4]

While for this puzzle it may seem obvious, the question remains for the other puzzles: Why are the other puzzles in this compendium also recursive?

Well, this drills down to the core of the matter. What do all these puzzles have in common, and how are they related to the "classic" Chinese Rings puzzle? A first insight might be to look at a solution method for Chinese Rings. The goal of this puzzle is to remove all the rings from the loop.

This can be established by a recursive algorithm with the following ideas:

move i-th ring off the loop:

Move ring i—1 (to the left of ring i) onto the loop

move all rings left of i—1 off the loop

then perform the movement of i-th ring

move i-th ring onto the loop:

Move ring i—1 (to the left of ring i) onto the loop

move all rings left of i—1 off the loop

then perform the movement of i-th ring

Here we see that for the movement of the i-th ring, preparation moves must be performed, and for these the algorithm can invoke itself recursively.
More details can be found in the book [3] and a more mathematical observations in the book [6].

Before the formalization of the recursive puzzles as above, a different notation was used in prior discussions: "n-ary Puzzles". This notation has its roots in well known puzzles: Chinese Rings, The Brain, SpinOut, The Key, and Binary Burr. These are typically called "binary" puzzles, because their "ring pieces" have two different states each, and their solution length function are asymptotically 2m. Ternary Burr, Tern Key, Crazy Elephant Dance are called "ternary" puzzles because they generalize the binary concepts to pieces having three different states. However, interestingly enough, their solution length functions are still asymptotically 2m, which is a justification for our structural definition. Why ternary puzzles can have 2m as solution length function is discussed below.

A mathematical argument for calling these puzzles "n-ary" is that their current state can be represented by an m-tuple of entries ranging in {0,...,n—1}. For example, the SpinOut starting configuration would be: (0,0,0,0,0,0,0) and the goal configuration: (1,1,1,1,1,1,1). A solution could then be described as a sequence of such tuples:

You may have noticed that In the definition of CR recursive puzzle we did not refer to the number of moves required to solve the puzzles, which is a central property in a different definition of the class of puzzles in this compendium (see discussion below). The number of moves seems related to the parameters in our definition, but no uniform relation has been determined for all the puzzles of this compendium, and this seems impossible, as we will see:
The key count is the number of moves that the special pieces (or "ring pieces") move during the solution process. Similar to approaches in Computer Science, we will sometimes not provide an exact function of the number of moves, but will use an approximate notation. In such cases the exact number of moves might not have been calculated yet, but by analogy one has a strong indication what it could be like.

This notation describes the asymptotic growth and we are only interested in the fastest growing elements of this number of moves function. Following are some examples for this notation:

Exact function

Approximation

Note

2m+m+3

Θ( 2m )

Linear and constant summands neglectable

2 · 5n

Θ( 5n )

Constant factor neglectable

2(m—1)

Θ( 2m )

2(m—1)= (1/2)·2m, constant factor neglectable

3 · (2m—1) — 2·m

Θ( 2m )

Combination of examples above

The Θ notation is taken from Computer Science and Mathematics, the formal definition and details are explained in [5].

Why do some CR recursive puzzles with n>2 have a solution with only ~2m moves, not ~nm?
CR recursive puzzles with solution length ~nm obviously use all different combinations of piece positions in their solution. To see this, just recall that the number of m-tuples over a set with n different values is exactly nm and so all these tuples occur in the description of the solution (see tuple representation above). So the puzzles in question that have solution length ~2m will not use all these tuples in their solution, but leave some out. For example, on the (shortest) solution path for the Ternary Burr, you will never find a configuration corresponding to tuple (1,0,0,0) -- this configuration is not needed. If you have this puzzle or the equivalent Crazy Elephant Dance, just try it out (with the lowest 4 elephants)! When you try the solution for on of these puzzles, you will find that directly before reaching this position, you will have the configuration (1,0,0,1) -- which is not part of the shortest solution either -- and this will be the only successor configuration after (1,0,0,1). Shortly said: only from (1,0,0,1) you reach (1,0,0,0), and your only option is going back to (1,0,0,0), and this is a detour way back from the solution from (0,0,0,0) to (2,0,0,0). Please see picture for these solution steps in the actual puzzle.

This effect occurs in several different puzzles, but what are the reasons for some puzzles to use all possible configurations, and for some others to omit configurations in their solution? Several reasons have been observed so far:

Condition for moves: Puzzles with all configurations (and hence solution length nm) have conditions that involve both lower and higher pieces. For an example see the Kugellager above. Others do only involve a part of the other pieces. The examples in this paragraph are of such nature. For Ternary Burr and Crazy Elephant Dance, it only matters which positions the lower pieces have. Pieces may be moved no matter in which positions the higher pieces currently are. This allows to bring a piece from position 0 to n—1 without touching the higher pieces at all. When solving Kugellager, you will notice that when moving the lowest piece later in the solution, you will have to move higher pieces before.

The second observation deals with Tower of Hanoi and Rudenko Clips. Both have the same general structure (3 positions) and equivalent rule:

Tower of Hanoi: bigger disc may not be on smaller one

Rudenko Clips: smaller clip may not be around bigger one

However, the first one has a solution length ~2m, while the Clips need ~3m moves. Here, the condition is the same, but there is an additional condition in Rudenko Clips: The three positions 0, 1, and 2 are in a row and a clip may only move between positions 0 and 1, if the stack of smaller clips is on 2, and a clip may only move between positions 1 and 2, if the stack of smaller clips is on 0. No direct move from 0 to 2 is possible for all clips except the biggest one. For Towers of Hanoi, we may use positions freely (only obeying the "bigger disc" rule). In the picture below, the red clip is not able to move from position 0 to 2 over the green one, while this would be the canonical next step in Tower of Hanoi. For a discussion on graph representations also leading to 2m and 3m as solution lengths, please see [4].

In the paragraphs above the relation between binary and ternary regarding the solution length has been discussed. There is also a binary representation that can be used for describing the solution of Tower of Hanoi, for details, please see [4].

Recently Goh Pit Khiam created a nice illustrated example of a variant of Tower of Hanoi: Linear Tower of Hanoi. In this variant, the three poles are in a line and a disc may only move from a pole to an adjacent pole. The obvious implications are that there are no direct moves between pole 0 and 2, and that a bigger disc may not move between poles 0 and 2, whenever there is a smaller one on pole 1. This makes it very similar to a Rudenko Clips (see previous section above). The less obvious implication is that the puzzle follows a ternary gray code. Please see [10] for the illustrated example of this puzzle, which also shows a nice ternary representation of Tower of Hanoi.

Our definition of CR recursive puzzle is based on the structure of the puzzle, which makes it (relatively) easy to spot if a puzzle belongs to this class. However, there is another different and commonly used definition of n-ary puzzles that first requires the puzzle to be analyzed and solved fully before it can be classified. Once one has determined that there are m special pieces and the solution length function is asymptotically equal to nm, it is classified as n-ary (in this solution length based notation). We are using the structural definition provided above (as it seems easier to apply it in most cases), but there might be some confusion. Some ternary puzzles (our definition) may have a solution with (asymptotic) length 2m, while the solution length function based definition would call them "binary" for this reason. One prominent example is the Ternary Burr by Pit Khiam Goh. This ternary (sic!) variation of the Binary Burr by Bill Cutler would be classified as binary following the solution length based definition.

The idea for this compendium dates back a few years, and in 2012 first steps were taken for implementation, starting to collect data, searching for new puzzles and determining a common property to create a formal definition.

I wish to thank Dan Feldman for big support in the creation of this compendium during many discussions, with research on certain puzzles and editorial work on this compendium. My thanks also go to Nick Baxter and Michel van Ipenburg, with whom I had some detail discussions about the compendium and some puzzles included.

Of course I do not own all the puzzles and need pictures of puzzles that I do not have, of course observing the copyright. Thank you for picture contributions or puzzle samples for taking pictures to: Dan Feldman, Jack Krijnen, Namick Salakhov, Rob Stegmann, Dirk Weber, Yvon Pelletier, Claus Wenicker, Allard Walker, Michel van Ipenburg, Nick Baxter, Robert Hilchie, Kevin Sadler, Jerry McFarland, Stephen Miller, Jeremy Rayner, James Dalgety, and Fredrik Stridsman. Thanks to Jan de Ruiter for pointing out the similarity between Quatro and Chinese Rings. Thanks to Pit Khiam Goh for some interesting discussions around BurrTools models of the puzzles and for confirming some puzzle entry details, and also nice illustrations of puzzles and puzzle examples. Many thanks to the designers and craftsmen who provided some of their puzzles for my collection or some detail descriptions of the puzzles.

Ingenious Rings: Many thanks to Wei Zhang, Peter Rasmussen and Nick Baxter for providing me material on these wire puzzles, of which I selected the ones that seem to fit the definition well. Beside the book [7] they also have a nice web site [8] and [9] on this topic. (see references above)

This compendium is not a static collection of puzzles, but a dynamic overview which will be updated when new puzzles, new details, pictures or references are available. If you would like to send me some feedback on the compendium, submit additional material or information to be added, or update some entries, please send a mail to me:

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Your feedback and contributions are welcome, so please do not hesitate. I am always interested to hear some interesting background stories about the puzzles.

If you are sending pictures for publication in the compendium, please clearly indicate that you are the holder of the rights on these pictures and that you allow the use in this compendium.